The observation of two PetaelectronVolt (PeV)-scale neutrino events reported by Ice Cube allows one to place constraints on Lorentz invariance violation (LIV) in the neutrino sector. After first arguing that at least one of the PetaelectronVolt IceCube events was of extragalactic origin, I derive an upper limit for the difference between putative superluminal neutrino and electron velocities of less than or equal to approximately 5.6 x 10(exp -19) in units where c = 1, confirming that the observed PetaelectronVolt neutrinos could have reached Earth from extragalactic sources. I further derive a new constraint on the superluminal electron velocity, obtained from the observation of synchrotron radiation from the Crab Nebula flare of September, 2010. The inference that the greater than 1 GigaelectronVolt gamma-rays from synchrotron emission in the flare were produced by electrons of energy up to approx. 5.1 PetaelectronVolt indicates the nonoccurrence of vacuum Cerenkov radiation by these electrons. This implies a new, strong constraint on superluminal electron velocities delta(sub e) less than or equal to approximately 5 x 10(exp -21). It immediately follows that one then obtains an upper limit on the superluminal neutrino velocity alone of delta(sub v) less than or equal to approximately 5.6 x 10(exp -19), many orders of magnitude better than the time-of-flight constraint from the SN1987A neutrino burst. However, if the electrons are subluminal the constraint on the absolute value of delta(sub e) less than or equal to approximately 8 x 10(exp -17), obtained from the Crab Nebula gamma-ray spectrum, places a weaker constraint on superluminal neutrino velocity of delta(sub v) less than or equal to approximately 8 x 10(exp -17).